Lagrange Points

In orbital dynamics, there are very strange things that can happen.  One of those is the idea of Lagrange Points.  When you have two bodies that are orbiting each other (like the Earth orbiting the Sun or the Moon orbiting the Earth), you can get points in which the forces cancel such that you can put satellites there and they will basically stay in the same spot (mostly).  There are five Lagrange points, as illustrated below. These are very strange and sort of unexpected!  I thought that I would explain how L1 and L2 are created and wave my hands for the others, since the physics is exactly the same.

The five Lagrange points between the Sun and Earth (from space.com)

To illustrate, we will consider the Earth-moon system, since that is a bit closer to home. Let’s start with the most basic thing: Earth’s gravity.  Earth has gravity that essentially goes outwards forever . Even Jupiter “feels” Earth’s gravity – it is just really weak.  Just to make sure that we are all on the same page, Earth’s gravity is towards Earth. 🙂


Let’s switch to the moon. When you are standing on the moon, you are always being pulled towards the moon, but if you look at this from the Earth’s perspective, this “towards the moon” can be either “towards the Earth” or “away from the Earth”, depending on which side of the moon you are on.  If you are on the side facing the Earth, the moon is pulling you away from the Earth.  If you are on the “Dark Side of the Moon” (i.e., the side of the moon facing away from the Earth), then the moon is actually pulling towards Earth.  Weird, right? Take a look at the plot:


So, if you next consider Earth’s gravity and the moon’s gravity together (plot below), you can see three things: (1) when you are close to the Earth, Earth’s gravity “wins”, and you are pulled towards Earth; (2) on the far side of the moon, Earth’s gravity and the moon’s gravity point in the same direction (towards the Earth), so they add together and you would be a little heavier on this side of the moon; and (3) when you are close to the moon, but on the Earth side of the moon, the moon pulls you away from the Earth and the Earth pulls you away from the moon, so you would weigh a bit less on this Earth side of the moon.  At some point between the Earth and the moon, there is a location in which Earth’s gravity (pulling you towards Earth) and the moon’s gravity (pulling you towards the moon) cancel each other and you have a gravitation null.


Solving for this gravitational null (see below) is a pretty standard high school physics problem.


Up until this point, we have only considered gravity.  This would be fine if Earth and the moon were fixed in place, instead of the moon orbiting around the Earth. But since this is not the case, we have to consider the orbital motion.  The moon goes around the Earth once every 28 days (roughly).  You can then ask the question about how much force is needed to keep the moon moving around in a circle instead of the moon flying off in a straight line.  This is the centripetal force.  You can experience this force if you go on a merry-go-round that is spinning fast. You have to hold on to something to keep from being flung off of it.  If you are standing at the exact center, then you don’t feel any force at all, but as you move further and further towards the edge, you end up feeling more and more force.  You should totally go and try this.

Well, the same force works in space.  If you have an object going around another object, it feels an outward force (well, the object wants to continue to move in a straight line, which we interpret as an outwardly directed force).  If we pretend that the moon is sitting on a gigantic merry-go-round at the position of the moon, and the merry-go-round is spinning exactly the same speed as the moon is going around the Earth (one revolution every ~28 days), the acceleration that you would feel at any point along the merry-go-round would be this:


Notice that the acceleration is always away from Earth, just like on the merry-go-round, you always feel a force pushing you away from the center.  Now, the Earth and the moon are not sitting on a gigantic merry-go-round, so this is really a thought experiment, but you get the idea.

This ends up being a third force/acceleration that is felt in the Earth-moon system, and so it needs to be included in all of our accelerations that we talked about earlier:


Notice that the centripetal acceleration is pretty much nothing compared to the Earth’s gravity until really close to the moon.  Is that a coincidence?  No!  What is the force that makes the moon orbit the Earth? Gravity balanced with centripetal acceleration!  If you trace the Earth’s gravity line (which turns into a dotted red line) and the centripetal acceleration line, they cross at the moon’s orbit (~60 Re)!  Physics!

But, we are really looking at the sum of the forces.  So, the centripetal acceleration becomes larger than all of the accelerations in the gravitational null point (since the gravity of the Earth and the gravity of the moon cancel so there is almost no acceleration there), and when you go on the other side of the moon and are far enough away, the centripetal acceleration becomes larger than the sum of the Earth and moon’s gravity.  Notice that the sum of the gravity is red (towards Earth), and the Centripetal acceleration is blue (away from Earth), which means that where they cross, the sum will be zero:


The above plot shows the sum of all three of the accelerations: gravity from the Earth, gravity from the moon, and centripetal acceleration. Now you can see that there are two areas where the three accelerations cancel each other out: around 52 Re away from the Earth and about 71 Re away from the Earth, both along the Earth-moon line.  These are the first two Lagrange points (L1 and L2).

These Lagrange points exist in any system where you have one body orbiting another.  You can look at this plot and think of the sun as being the main body, and the Earth as being the second body, so that L1 is between the sun and Earth, and L2 is away from the sun on the dark side of the Earth. These Lagrange points are very useful, since we can place satellites near them so that they can look at the sun all of the time (this is useful for solar physics missions), or look back at the dayside of the Earth all of the time (for climate and weather missions), or be in the shade of the Earth all of the time (well, no missions really want this, since they need solar power to run, but they do want to be far away from the Earth and not have the sun be too large in their view).

There are three other Lagrange points that I have not discussed (as shown in the top picture as L3, L4, and L5).  All of these points result from the balance between the three accelerations, just like L1 and L2. L3 is very cool because it is on the opposite side of the sun, and we can’t really observe it.  Some people would argue that there could be a mirror Earth at L3, which is pretty funny.  We would be able to tell though, since the hidden planet would change the orbits of Venus and Mercury.  But, can we really trust science? 🙂

L4 and L5 are strange because they are in Earth’s orbital path around the sun, but about 60 degree behind and in front of the Earth.  The math is a tiny bot more complicated, since you have to consider two dimensions, but the concept is exactly the same. These two Lagrange points are really interesting, since you can observe the sun from unique vantages.